Parallel and Perpendicular Lines
Every question with a fully worked solution.
- Parallel and Perpendicular Lines - Video - Parallel, Intersection and Perpendicular Line Watch
Theory
Parallel lines are straight lines on a flat surface that never meet. Perpendicular lines meet at a right angle (\(90^\circ\)). This Year 7 maths page explains the symbols \(\parallel\) and \(\perp\), the marks used on diagrams, transversals, and parallel and perpendicular sides of shapes and lines on the number plane.
When two lines cross at one point, they intersect. Line \(CD\) intersects line \(AB\) at the point where they cross.
Parallel lines go in the same direction and never meet, however far they are extended. Matching arrow marks show parallel lines. \(\parallel\) means “is parallel to”: \(AB \parallel CD\).
Perpendicular lines meet at \(90^\circ\). A small square shows the right angle. \(\perp\) means “is perpendicular to”: \(AB \perp CD\).
A transversal is a line that crosses two or more other lines.
Useful facts. Two lines perpendicular to the same line are parallel. Two lines parallel to the same line are parallel to each other. A line perpendicular to one of a set of parallel lines is perpendicular to all of them. The diagonals of a square or a rhombus are perpendicular.
Marks and symbols:
| Mark or symbol | Meaning |
|---|---|
| matching arrow marks on two lines | the lines are parallel |
| a small square | a right angle, so perpendicular |
| matching tick marks on two intervals | equal lengths |
| \(AB \parallel CD\) | \(AB\) is parallel to \(CD\) |
| \(AB \perp CD\) | \(AB\) is perpendicular to \(CD\) |
How to describe two lines
- Do they meet (or would they, if extended)? If not, and they have arrow marks, they are parallel.
- If they meet with a right-angle mark or a \(90^\circ\) label, they are perpendicular.
- Otherwise they are intersecting but not perpendicular.
- Write the answer with \(\parallel\) or \(\perp\) where you can.
(i) a pair of parallel lines
(ii) the line that is perpendicular to the parallel lines
(i) \(AB\) and \(CD\) never meet: they are parallel.
(ii) \(EH\) meets \(CD\) at a right angle, so it is perpendicular to both parallel lines.
(i) \(AB \parallel CD\)
(ii) \(EH \perp AB\) and \(EH \perp CD\)
(i) Which side is parallel to \(AD\)?
(ii) Name the two sides perpendicular to \(AB\).
(iii) Is \(AC \perp AB\)?
(i) Opposite sides are parallel.
(ii) Sides that meet make a right angle.
(iii) The right-angle mark at \(A\) is for \(\angle DAB\), not \(\angle CAB\), which is only part of that corner.
(i) \(BC\)
(ii) \(AD \perp AB\) and \(BC \perp AB\)
(iii) No
(i) Describe \(AB\) and \(CD\) with a symbol.
(ii) \(E\) is \((5, 1)\). Give a point \(F\) so that \(EF \parallel CD\).
(i) \(A\) and \(B\) have the same \(y\), so \(AB\) is horizontal. \(C\) and \(D\) have the same \(x\), so \(CD\) is vertical.
(ii) \(EF\) must be vertical, like \(CD\), so \(F\) has \(x = 5\).
(i) \(AB \perp CD\)
(ii) For example \(F(5, 4)\): any \((5, y)\) with \(y \ne 1\).
(i) the value of \(x\)
(ii) the size of \(\angle AOE\)
(i) \(AB \perp CD\), so \(\angle BOD = 90^\circ\). Angles in a right angle add to \(90^\circ\).
(ii) Angles on a straight line add to \(180^\circ\).
| \(x + 2x\) | \(=\) | \(90\) |
| \(x\) | \(=\) | \(30\) |
| \(\angle AOE\) | \(=\) | \(180 - 30 = 150\) |
(i) \(x = 30\)
(ii) \(150^\circ\)
(i) What is true about \(c\) and \(b\)?
(ii) What is true about \(d\) and \(a\)?
(i) A line perpendicular to one of two parallel lines is perpendicular to the other one too.
(ii) \(a\) and \(d\) are both perpendicular to \(c\), and two lines perpendicular to the same line are parallel.
(i) \(c \perp b\)
(ii) \(d \parallel a\)
Common pitfalls
Frequently asked questions
What are parallel lines?
Straight lines on a flat surface that never meet.
What are perpendicular lines?
Lines that meet at a right angle, \(90^\circ\).
What does AB ∥ CD mean?
\(AB\) is parallel to \(CD\).
What does AB ⊥ CD mean?
\(AB\) is perpendicular to \(CD\).
What is a transversal?
A line that crosses two or more other lines.